Rubio: America has a 'violence problem'

4/14/13 9:53 AM EDT

"Guns are what people are using," the Republican senator from Florida said Sunday on NBC's "Meet the Press." "But violence is our problem."

Asked by host David Gregory how additional checks would undermine the Second Amendment, Rubio says it's false to assume that "criminals...will no longer get guns because they have to undergo a background check."

And he rejected host Gregory's assessment that he had to be careful not to stray too far from his party on guns, saying that his position remains unchanged and citing the Second Amendment.

"I think this is a missed opportunity to have an honest and open conversation in this country about why these horrifying things are happening, not simply what they're using to carry this out, but why are people doing this to begin with?" Rubio said.

Rubio said he hasn't fully read the Manchin-Toomey gun compromise, but he said he'd consider legislation "that keeps guns out of the hands of criminals but protects the right of law-abiding citizens to possess them."

Earlier on CNN's "State of the Union," Sen. Joe Manchin (D-W.Va.) said "law-abiding" gun owners would like the legislation.